Question

Explain how a directional derivative is formed from the two partial derivatives \(f_{x}\) and \(f_{y}\).

Short answer

Answer: Partial derivatives help form directional derivatives by being the components of the gradient vector of the function. The gradient vector consists of the partial derivatives with respect to each variable. The directional derivative is then computed as the dot product of the gradient vector and the given unit vector which represents the specific direction. This way, partial derivatives are combined with the components of the unit vector to find the rate of change of the function in the desired direction.

Step-by-step solution

Understand partial derivatives

Partial derivatives are derivatives of a multivariable function with respect to one variable, while holding all the other variables constant. They represent the rate of change of the function in the direction of the chosen variable. In our case, we have a function of two variables, \(f(x, y)\), and its partial derivatives with respect to \(x\) and \(y\) are denoted as \(f_{x}\) and \(f_{y}\), respectively.

Understand directional derivatives

A directional derivative is a derivative of a multivariable function in the direction of a given vector. It gives us the rate of change of the function in that specific direction. The directional derivative of \(f(x, y)\) in the direction of the unit vector \(\vec{u}=(u_1, u_2)\) is denoted as \(\nabla_{\vec{u}}f(x, y)\).

Link partial derivatives to directional derivatives

To form the directional derivative from the partial derivatives, we use the gradient of the function. The gradient of a function \(f(x,y)\) is a vector formed by its partial derivatives: \(\nabla f(x, y) = (f_{x}(x, y), f_{y}(x, y))\). Now, to find the directional derivative \(\nabla_{\vec{u}}f(x, y)\), we take the dot product of the gradient vector with the given unit vector \(\vec{u}=(u_1, u_2)\): \(\nabla_{\vec{u}}f(x, y) = \nabla f(x, y) \cdot \vec{u} = (f_{x}(x, y), f_{y}(x, y)) \cdot (u_1, u_2)\)

Compute the directional derivative

The dot product is computed as follows: \(\nabla_{\vec{u}}f(x, y) = f_{x}(x, y)u_1 + f_{y}(x, y)u_2\) So, the directional derivative is formed from the two partial derivatives \(f_{x}\) and \(f_{y}\), by taking their linear combination with the components of the given unit vector as coefficients.

Key concepts

Partial Derivatives

To thoroughly understand partial derivatives, picture a surface that represents a function with two inputs, such as elevation across different points on a map. A partial derivative, then, is like looking at only one slice of that map—either north-south or east-west—and showing how steep the hill is if you walked in that single direction. Mathematically, for a function like f(x, y), the notation f_x denotes the partial derivative with respect to x, where you consider changes in f only when you move along the x-direction, keeping y constant. Similarly, f_y is the rate of change of f along the y-direction, fixing x.

Visualizing these rates of change is critical, as they are foundational for more advanced concepts in multivariable calculus. When computing them, you're effectively asking: 'If I alter just one variable slightly, how much does my function change?' If you imagine a three-dimensional graph, partial derivatives give you the slope of the tangent line on the function's curve at a specific point when projected onto the plane of the variable you're differentiating with respect to.

Multivariable Calculus

Stepping into the realm of multivariable calculus is like diving into a world where you can explore landscapes beyond the flatland of single-variable functions. Here, functions can have two or more inputs—think of f(x, y) as not just a line, but a surface that can bend, dip, and peak in space. It's a complex dance of interaction between these variables that can impact the outcome. You're not just looking at how things change in one direction, but in an entire space, with multiple paths to consider.

In this broader picture, differential calculus focuses on how functions change over tiny intervals, and you use partial derivatives as your tools to paint a detailed picture of those changes. You're not just learning about the rate of change in one path but learning to navigate an entire terrain of possibilities. This is not just mathematics – it's a way of understanding how multi-faceted systems behave in our world, from the physics of waves and heat to the dynamics of economies.

Gradient Vector

Imagine the gradient vector as a compass that tells you the steepest way up a mountain. Unlike a regular compass that points north, the gradient vector points towards the direction of the greatest rate of increase of some function—such as altitude on our mountain analogy. Symbolically, if you have a function f(x, y), its gradient, symbolized by ∇f, is a vector made up of all the partial derivatives (f_x, f_y).

So what's this got to do with climbing mountains or understanding functions? Well, if you want to ascend most efficiently, you follow the gradient vector's direction. Similarly, in mathematics, if you want to find out how fast the function is increasing and in which direction, you use the gradient. It's like telling you both the slope of the steepest path and the direction you need to take that path—summarizing how the function changes in all directions at a point into a single vector with both magnitude and direction.